ALG 11

THIS WEEK: We explore the phenomenon of Letter as Object, something we met briefly in ALG 9C, ie we look at contexts where a letter represents a pure number (of objects) or a quantity (the price or length or mass or some other numerical quality of an object), but where there might be a temptation to use the letter as a shorthand for the object itself (as in 'a stands for apple, b stands for banana' - the classic fruit salad algebra). We will see that sometimes one can slide harmlessly between letter as object and letter as quantity, but that sometimes it leads to a completely fake algebra.
MONDAY: In this pair of tasks m stands for a numerical quality (mass) in the first task and for a pure number in the second. Though the answer is 5m in both cases, we have found that the second kind of task is substantially more demanding than the first - does that hold for your students?

In the first task, it is easy to interpret m as simply standing for mints rather than the mass of a mint, but still come up with the correct expression, 5m, albeit misinterpreted as 5 mints. In the second task students have to cope with the idea that m is most definitely a number, but one whose value we don't know, so that it is not possible to arrive at a specific numerical answer.
TUESDAY: Careful! Needles and pins. Because of all my pride, the tears I gotta hide... This turns out to be quite a Searching task, though let's hope it doesn't quite lead to tears!

The task involves an expression in which the letters stand for numbers but where the temptation to treat them as objects is very strong (for us as well as our students) and where this leads to a complete misinterpretation of the expression.  
Why can't I stop
And tell myself I'm wrong
I'm wrong, so wrong
Why can't I stand up
And tell myself I'm strong
....
The extract below is from my article Object lessons in algebra? that appeared in Mathematics Teaching 98 in 1982. Still worth reading!

 

Here is another variant of the task, but where we are asked to write a relation rather than interpret an expression. It is again quite challenging.

WEDNESDAY: In this task, the misinterpretation that results from treating the letters as objects is not as jarring... The given expression stands for the cost (in number of pence) of 5 bananas and 2 coconuts, so it is wrong to simply translate it as 5 bananas and 2 coconuts. Nevertheless this translation does make some kind of sense - the story is about 5 bananas and 2 coconuts, whereas the corresponding interpretation of the expression 2p + 5n in ALG 11B, as 2 pins and 5 needles, does not fit that story at all.

It is interesting to consider the alternative version of task 11C shown below.

Here students are quite likely to come up with the right expression, though not necessarily for the right reason - we can't be sure whether students who write 5b + 2c fully realise that this represents a number of pence and that it is not simply telling us about the number of bananas and coconuts bought. So this alternative version is not as useful as the original for revealing students' thinking. The task below, which appeared recently on Twitter, is even less effective as a diagnostic tool: here the correct option (B, I assume) can be chosen by simply treating it as an abbreviation of the verbal statements, ie by reading 5t + 2c = 3.70 as 5 teas and 2 coffees cost 3.70 (pounds), rather than appreciating that 5t + 2c actually represents the number of £s spent on the drinks.

THURSDAY: We present the first of two tasks involving sets of coloured rods, which we ask students to symbolise in different ways.

In a sense, this task is quite easy - we can get it right by interpreting the letters correctly, leading to 3g = b (on the basis that 3 times the length g equals the length b), but we can also get it right by writing 3g = b as shorthand for 3 green rods make a blue rod, where g and b are perhaps being thought of merely as shortened names for the green and blue rods, and not necessarily as symbols for their lengths. Thus, in a task like this, students can appear to be thinking algebraically when perhaps they're not.
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FRIDAY: Here we meet the same rows of green rods and blue rods, but this time we relate the number of green rods and blue rods, not their lengths.

In this task, we can feel a strong pull towards writing 3g = b again, on the basis that 3 green rods make a blue rod, or there are 3 green rods for every blue rod. However, we need to fix firmly on the fact that in this version of the rods task, g and b are defined as numbers of rods, for example 3 and 1, or 6 and 2, or 9 and 3, etc. So g is 3 times b, ie g = 3b. This can feel counter-intuitive as it doesn't map onto our verbal descriptions of the situation.